Tensor product of von Neumann algebras Hi everyone,
I currently encountered some difficulties on my research related to operator algebras and I wondered if by any chance someone could find my question quite trivial. Here is the context: given a von Neumann algebra $A$, I know that we can construct the so-called von Neumann tensor product $A\otimes A$ of $A$, providing us with a natural completion of the algebraic tensor product of $A$. My question is then the following: can we continuously extend the multiplication map $u\otimes v \to uv$ (defined on the algebraic tensor product) to a linear map defined on the whole product $A\otimes A$? In other words, is this multiplication map continuous with respect to the topology of $A\otimes A$? If not, what if $A$ is endowed with a faithful trace in addition? These questions may be obvious for some operator algebras specialists, but I can’t find any paper/book on this specific issue…
Thank you for any help.
 A: The answer to your question is No (beyond matrix cases i.e. finite dimensional von Neumann algebras). Even the von Neumann tensor product of one abelian von Neumann algebra, which is indeed $L^\infty(X,\mu)$, does not have this property. 
Let 
$$
m: M \odot M \rightarrow M
$$
be the multiplication on the algebraic tensor product $\odot$ of a von Neumann algebra, that is  $m(v\otimes u) = uv$ for $u,v \in M$.
There is an appropriate tensor product on $M\odot M$ which  is suitable for linearizing multiplication map (weak$^*$ continuity) $m$ so-called   the normal Haagerup  tensor product, usually denoted by $M \otimes^{\sigma h} M$ (introduced initially in [1]). To see the proof of this fact and learn more about this tensor product look at [2].
References. 
[1] E. Effros and A. Kishimoto, Module maps and Hochschild-Johnson cohomology, Indiana Univ. Math. J. 36(1987), 257–276.
[2] E. G. Effros and Z.-J. Ruan, Operator spaces tensor products and Hopf convolution algebras, J. Operator Theory 50 (2003) 131–156.
